PERIODIC FLAT MODULES, AND FLAT MODULES FOR FINITE GROUPS
PERIODIC FLAT MODULES, AND FLAT MODULES FOR FINITE GROUPS
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DOI：
10.2140/pjm.2000.196.45
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发表时间：
2000-11
影响因子：
0.6
通讯作者：
D. Benson;K. Goodearl
中科院分区：
文献类型：
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作者：
D. Benson;K. Goodearl
若R是系数环，G是有限群，则投射为R-模的平坦RG-模必然投射为RG-模。更一般地，如果H是任意群Γ中的有限指数子群，则投射为RH-模的平坦RΓ模必然投射为RΓ-模。这是从第一个定理推广到强G-分次环上的模。这些结果是利用以下关于任意环S上平坦模的定理证明的：如果平坦S-模M位于短正合序列0 → M → P → M → 0中，且P是投射的，则M是投射的。还证明了有限群的群环上的平坦模和投射模的其他一些性质，包括模素的约化。
If R is a ring of coefficients and G a finite group, then a flat RG-module which is projective as an R-module is necessarily projective as an RG-module. More generally, if H is a subgroup of finite index in an arbitrary group Γ, then a flat RΓmodule which is projective as an RH-module is necessarily projective as an RΓ-module. This follows from a generalization of the first theorem to modules over strongly G-graded rings. These results are proved using the following theorem about flat modules over an arbitrary ring S: If a flat S-module M sits in a short exact sequence 0 → M → P → M → 0 with P projective, then M is projective. Some other properties of flat and projective modules over group rings of finite groups, involving reduction modulo primes, are also proved.